Let ˆθn be an estimator of a real-valued parameter θ such that XXXXXXXXXXholds for any θ and let Vˆn be a consistent estimator of Vn. Suppose that Vn → 0. (a) Show that the test with rejection region...

Let ˆθn be an estimator of a real-valued parameter θ such that (6.92) holds for any θ and let Vˆn be a consistent estimator of Vn. Suppose that Vn → 0.

(a) Show that the test with rejection region Vˆ −1/2 n (ˆθn−θ0) > z1−α is a consistent asymptotic test for testing H0 : θ ≤ θ0 versus H1 : θ > θ0.

(b) Apply the result in (a) to show that the one-sample one-sided t-test in §6.2.3 is a consistent asymptotic test.